3.675 \(\int e^{-4 x} (e^x+e^{2 x}+e^{3 x}) \, dx\)

Optimal. Leaf size=26 \[ -\frac{1}{3} e^{-3 x}-\frac{e^{-2 x}}{2}-e^{-x} \]

[Out]

-1/(3*E^(3*x)) - 1/(2*E^(2*x)) - E^(-x)

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Rubi [A]  time = 0.0220473, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2282, 14} \[ -\frac{1}{3} e^{-3 x}-\frac{e^{-2 x}}{2}-e^{-x} \]

Antiderivative was successfully verified.

[In]

Int[(E^x + E^(2*x) + E^(3*x))/E^(4*x),x]

[Out]

-1/(3*E^(3*x)) - 1/(2*E^(2*x)) - E^(-x)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int e^{-4 x} \left (e^x+e^{2 x}+e^{3 x}\right ) \, dx &=\operatorname{Subst}\left (\int \frac{1+x+x^2}{x^4} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{x^4}+\frac{1}{x^3}+\frac{1}{x^2}\right ) \, dx,x,e^x\right )\\ &=-\frac{1}{3} e^{-3 x}-\frac{e^{-2 x}}{2}-e^{-x}\\ \end{align*}

Mathematica [A]  time = 0.0111108, size = 23, normalized size = 0.88 \[ -\frac{1}{6} e^{-3 x} \left (3 e^x+6 e^{2 x}+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x + E^(2*x) + E^(3*x))/E^(4*x),x]

[Out]

-(2 + 3*E^x + 6*E^(2*x))/(6*E^(3*x))

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Maple [A]  time = 0.021, size = 20, normalized size = 0.8 \begin{align*} -{\frac{1}{3\, \left ({{\rm e}^{x}} \right ) ^{3}}}-{\frac{1}{2\, \left ({{\rm e}^{x}} \right ) ^{2}}}- \left ({{\rm e}^{x}} \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+exp(2*x)+exp(3*x))/exp(4*x),x)

[Out]

-1/3/exp(x)^3-1/2/exp(x)^2-1/exp(x)

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Maxima [A]  time = 0.976115, size = 26, normalized size = 1. \begin{align*} -e^{\left (-x\right )} - \frac{1}{2} \, e^{\left (-2 \, x\right )} - \frac{1}{3} \, e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+exp(2*x)+exp(3*x))/exp(4*x),x, algorithm="maxima")

[Out]

-e^(-x) - 1/2*e^(-2*x) - 1/3*e^(-3*x)

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Fricas [A]  time = 0.759663, size = 53, normalized size = 2.04 \begin{align*} -\frac{1}{6} \,{\left (6 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+exp(2*x)+exp(3*x))/exp(4*x),x, algorithm="fricas")

[Out]

-1/6*(6*e^(2*x) + 3*e^x + 2)*e^(-3*x)

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Sympy [A]  time = 0.104578, size = 22, normalized size = 0.85 \begin{align*} - e^{- x} - \frac{e^{- 2 x}}{2} - \frac{e^{- 3 x}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+exp(2*x)+exp(3*x))/exp(4*x),x)

[Out]

-exp(-x) - exp(-2*x)/2 - exp(-3*x)/3

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Giac [A]  time = 1.21308, size = 24, normalized size = 0.92 \begin{align*} -\frac{1}{6} \,{\left (6 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+exp(2*x)+exp(3*x))/exp(4*x),x, algorithm="giac")

[Out]

-1/6*(6*e^(2*x) + 3*e^x + 2)*e^(-3*x)